积分型样条函数逼近新理论、新方法及应用研究

项目名称： 积分型样条函数逼近新理论、新方法及应用研究

项目编号： No.11501533

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 郎丰贡

作者单位： 中国海洋大学

项目金额： 18万元

中文摘要： 根据某一元或多元函数的积分值信息逼近此函数及其各阶导数或偏导数是一类基于线性泛函信息的函数逼近问题。本项目研究积分值型样条函数逼近问题的新理论、新方法及其应用，此方面已有的工作具有若干不足与局限，尚不够深入与完善。在本项目中，主要以样条函数为工具，充分利用给定的积分值数据，重点研究高效积分值型样条函数插值和拟插值以及近似插值和近似拟插值的逼近理论方法、误差估计、数值算法及实际应用。主要包括以下工作:（1）研究适用于一般非周期函数且不需要任何边界条件或参数的理论方法；（2）研究能够逼近函数本身以及若干阶导数或偏导数的高精度逼近理论方法；（3）深入开展研究多元积分型样条函数理论、性质、方法；（4）设计简洁高效、易于实现、计算量低的新算法；（5）研究新理论新方法在数值分析、数理统计、环境科学等领域的应用。

中文关键词： 积分型样条函数；积分插值；积分拟插值；多元样条

英文摘要： The purpose of this project is to research the integro spline approximation theory, methods and applications. The previous studies on this topic are very limited and need to be improved greatly. We will apply spline functions to study some new highly effective numerical methods for approximating a univariate or multivarite function and its derivatives or partial derivatives from the given integral values of successive subintervals or subdomains. Essentially, it is an approximation problem based on linear functionals. We will analyze the approximation theory and develop the numerical algorithms for the so-called integro interpolating splines, approximate integro interpolating splines, integro quasi-interpolating splines and approximate integro quasi-interpolating splines. Moreover, we will also study their practical applications in many fields. Our new theory and methods are greatly improved and have many advantages over the existing ones. (1) The new theory and methods will be applicable to non-periodic functions and will not need any additional boundary conditions or parameters; (2) The new integro splines will be able to approximate the function and its derivatives or partial derivatives with higher approximation ability; (3) The theory and methods of multivariate integro splines will be well studied; (4) The new algorithms will have lower computational complexity and be easy to implement; (5) The applications of our new theory and methods in numerical analysis, mathematical statistics and environmental science will be studied.

英文关键词： integro spline function;integro interpolation;integro quasi-interpolation;multivariate spline